The Yokonuma-Hecke algebras and the HOMFLYPT polynomial. (English) Zbl 1295.57015
The Yokonuma-Hecke algebras \(Y_{d,n}(u)\) were introduced by T. Yokonuma [C. R. Acad. Sci., Paris, Sér. A 264, 344–347 (1967; Zbl 0225.20027)] in the context of Chevalley groups, as generalizations of the Iwahori-Hecke algebras. In [J. Knot Theory Ramifications 13, No. 1, 25–39 (2004; Zbl 1100.20008)], J. Juyumaya constructed a unique linear Markov trace \(tr\) on the algebras \(Y_{d,n}(u)\), depending on d parameters. The trace \(tr\) was used sub-sequently by one of the authors in [J. Juyumaya and S. Lambropoulou, Adv. Math. 234, 149–191 (2013; Zbl 1266.57011)] for defining isotopy invariants for framed knots. The authors compare the invariants for classical knots and links defined using the Juyumaya trace on the Yokonuma-Hecke algebras with the HOMFLYPT polynomial. They show that these invariants do not coincide with the HOMFLYPT except in a few trivial cases.
Reviewer: Claus Ernst (Bowling Green)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 20F36 | Braid groups; Artin groups |
| 20C08 | Hecke algebras and their representations |
References:
| [1] | Geck M., London Mathematicsl Society Monographs, New Series 21, in: Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras (2000) |
| [2] | DOI: 10.2307/1971403 · Zbl 0631.57005 · doi:10.2307/1971403 |
| [3] | DOI: 10.1142/S0218216504003020 · Zbl 1100.20008 · doi:10.1142/S0218216504003020 |
| [4] | DOI: 10.1016/j.topol.2007.01.010 · Zbl 1165.57007 · doi:10.1016/j.topol.2007.01.010 |
| [5] | DOI: 10.1016/j.aim.2012.10.011 · Zbl 1266.57011 · doi:10.1016/j.aim.2012.10.011 |
| [6] | DOI: 10.1142/S0218216509007324 · Zbl 1188.57010 · doi:10.1142/S0218216509007324 |
| [7] | DOI: 10.1007/978-3-642-15637-3_5 · Zbl 1222.57010 · doi:10.1007/978-3-642-15637-3_5 |
| [8] | Yokonuma T., C.R. Math. Acad. Sci. Paris 264 pp 344– |
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